This is exactly what thought is, and perhaps consciousness itself. When you look around a room, you see blues and grays, flats and curves, hards and softs; the moment something enters your thoughts, it is labeled; it is impossible to do otherwise (you can't think of something without thinking about it and making associations). You are looking at language. The mind works by association and language is merely association. When I look at a glass of water, much more than "glass of water" enters my mind. I question whether I'm thirsty or not; I wonder if someone will spill it; I get the urge to take a piss. This is difficult to grasp, but here is a simple equation: language = symbolism = thought. Which brings me to the point:

You are language. By you, I mean the voice in your head that right now is reading these words off the screen. Look away for a moment and words will continue to flow through your mind. You can't help it. Where are the spaces between thoughts? At this level, you are not in control of your own mind. This is probably the first thing you learned if you've ever tried meditation.

"A system of symbols developed to communicate
ideas between intelligent entities."

can be abbreviated somewhat.

The term symbol presupposes use for communication. Communication presupposes intelligence, the ability to understand the communication. I think the term idea is distracting; as long as information is transmitted, I would consider it language. When two Internet computers speak a TCP/IP protocol, they are using a language, as far as I'm concerned.

This leaves the following definition:

"A system of symbols."

A symbol is something that signifies something: it stands for something else not merely through its own intrinsic properties, but at least partially through convention, an agreement between the symbol's users.

If further qualification is necessary, I would say it's in the use of the word "system". In language two forms of systematicity play together: paradigmatic and syntagmatic. A paradigm is a set of items that alternate in a systematic way; a syntagm is a set of items that combine in a systematic way. The way items can be combined is called the syntax of a language.

In a purely paradigmatic system, there is no meaning to the combination of items; just the choice of item carries meaning. (This is, for instance, the case with traffic signs; and it is also apparently the level at which we know chimpanzees can understand language: they can agree on which signs to use, but not in which order.) If we require syntax, it becomes questionable whether the traffic signs form a language: is there any meaning in combinations there? The word 'system' by itself is not very clear on whether syntax is presupposed or not. (As to chimpanzees, it has been claimed that some of them do use different combinations of the same symbols to mean different things.)

Why do I bother? Surely it's all a matter of definition? Well, yes, but it helps in discussing such issues as whether language is what distinguishes humans from the other animals. When someone makes that claim, do they purely mean reference, the power to refer to something? Then pointing, gesturing and other forms of body language would already qualify. Do they refer to the use of symbols, the power to refer to something through an established convention of use? Or do they refer to syntax, the ability to express meaning in the way utterances are combined? Many animals can take the first two hurdles, but the last step is pretty much reserved to humans.

Third, while language in its widest possible use is very important to me, I'd like to stress that I'm much more.
The intelligence I use in driving cars or dealing with other people has little or nothing to do with language.

Even in communication, not all is language. Body language, for instance, can hardly be called symbolic in nature.